Problems

{ } 3.10.
Problems for section 3.2.
3.1.
Use Ampère's law to find the magnetic field at a radial distance r from a long straight current-carrying wire.

3.2.
The Joule heating law says that the power dissipated by a current-carrying object is



where V is the voltage, I is the current, and R is the resistance. Furthermore, for small temperature changes , the fractional change of length of a solid obeys



where is the linear coefficient of thermal expansion of the material. Discuss the relevance of these two physical laws on the string apparatus.

Section 3.3.
3.3.
From equation (3.3), show that the location of the two equilibrium points is approximately given by .

3.4.
Solve the differential equation and graph x(t) versus t for a few values of . Why is called the damping coefficient?

3.5.
Solve the differential equation , , . Draw a plot of for a fixed .

3.6.
Verify that the variables in equation (3.11) are dimensionless.

3.7.
Verify that equation (3.13) is a solution to equation (3.7) with and K = 0 (discard second-order terms in ). Draw the solution, , in the x-y plane.

3.8.
Derive equation (3.2) from Figure 3.3. Hint: To account for the factor of 2 realize that each spring makes a separate contribution to the restoring force.

Section 3.4.
3.9.
(a) Show that the equation of motion for a simple pendulum  in dimensionless variables is



(b) Write this as a first-order system with . Using equation (3.18), find the potential energy function and a few integral curves for a pendulum.

(c) Show that the pendulum has equilibrium points at (0,0) and and discuss the stability of these fixed points by relating them to the configurations of the physical pendulum. Are there any saddle points? Are there any centers?

(d) Draw a schematic of the phase plane. Identify the separatrix in this phase portrait. Orbits inside the separatrix are called oscillations. Why? Orbits outside the separatrix are called rotations. Why?

(e) Add a dissipative term ( ) to get a damped pendulum and discuss how this changes the phase portrait. In particular, discuss the relation of the insets and outsets of the equilibrium points with the basins of attraction. Are there any attractors? Are there any repellers?

(f) Now add a forcing term and write the differential equations for the system in the extended phase space , where . Define a global cross section for the forced damped pendulum  (see eq. (3.29)).

Section 3.5.
3.10.
Verify that equation (3.32) is a solution to equation (3.31) (discard higher-order terms in ).

3.11.
Plot the linear response curve (eq. (3.34)) for a few representative values of F and .

3.12.
This exercise illustrates the method of harmonic balance . Assume an approximate solution of the form



to the Duffing equation (3.23).

(a) Substitute into equation (3.23) and equate terms containing and separately to zero.

(b) Show that



where



(c) Show that



(d) Plot the response ( versus ) for , , and several values of F. Plot the amplitude characteristic ( versus F) for , , and . Indicate the unstable solution with a dashed line. Hint: It is easier to plot as the independent variable.

Section 3.7.
3.13.
Verify equations (3.52-3.54) for the free whirling motions of a string.

3.14.
Write a program to iterate the circle map (eq. (3.57)) and explore its solutions for different values of K and .

Section 3.8.
3.15.
For a time series , in which the are sampled at evenly spaced times, write a transformation between the phase space variable and the embedded phase space variable , with r = 1. Why is the embedded phase space rotated like it is in Figure 3.24? Why does one usually choose r ;SPMgt; 1?

3.16.
Write a program based on equations (3.67-3.71) to calculate the discrete Fourier amplitude coefficients and the power spectrum (eq., (3.72)) for a discrete time series. Test the program on some sample functions for periodic motion (e.g., , and quasiperiodic motions (e.g., .